In this paper I will defend a logical conception of refutations and fallacies against objections that are meant to show that a dialectical conception of refutations or fallacies is necessary. I will show that there is only one dialectical concept—not that of a thesis, as those favouring a dialectical analysis argue, but that of a concession—that may need to be added to a logical conception for such a conception to be adequate.

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk's logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.

A 'self-refutation argument' is any argument which aims at showing that a certain thesis is self-refuting. This study was the first book-length treatment of ancient self-refutation and provides a unified account of what is distinctive in the ancient approach to the self-refutation argument, on the basis of close philological, logical and historical analysis of a variety of sources. It examines the logic, force and prospects of this original style of argumentation within the context of ancient philosophical (...) debates, dispelling various misconceptions concerning its nature and purpose and elucidating some important differences which exist both within the ancient approach to self-refutation and between that approach, as a whole, and some modern counterparts of it. In providing a comprehensive account of ancient self-refutation, the book advances our understanding of influential and debated texts and arguments from philosophers like Democritus, Plato, Aristotle, Epicurus, the Stoics, the Academic sceptics, the Pyrrhonists and Augustine. (shrink)

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk''s logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.

This study introduces refutation-aware Gentzen-style sequent calculi and Kripke-style semantics for propositional until-free linear-time temporal logic. The sequent calculi and semantics are constructed on the basis of the refutation-aware setting for Nelson’s paraconsistent logic. The cut-elimination and completeness theorems for the proposed sequent calculi and semantics are proven.

Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.

The goal of this paper is to analyse Implicational Relevance Logic from the point of view of refutability. We also correct an inaccuracy in our paper “The RM paraconsistent refutation system”.

Axiomatic proof/refutation systems for the paraconsistent modal logics: KN4 and KN4.D are presented. The completeness proofs boil down to showing that every sequent is either provable or refutable. By constructing finite tree-type countermodels from refutations, the refined characterizations of these logics by classes of finite tree-type frames are established. The axiom systems also provide decision procedures for these logics.

The purpose of the paper is to study syntactic refutation systems as a way of characterizing normal modal propositional logics. In particular it is shown that there is a decidable modal logic without the finite model property that has a simple finite refutation system.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

In the paper rejection systems for a system of nonsense-logic are investigated. The first rejection system consists of four rejected axioms and only one rejection rule  the rule of rejection by detachment. The second one consists of one rejected axiom and two rejection rules: the rule of rejection by detachment and the rule of rejection by substitution. The aim of the paper is to present also a proof of Ł-decidability for the considered systems.

We offer a critical overview of two sorts of proof systems that may be said to characterize classical propositional logic indirectly (and non-standardly): refutation systems, which prove sound and complete with respect to classical contradictions, and rejection systems, which prove sound and complete with respect to the larger set of all classical non-tautologies. Systems of the latter sort are especially interesting, as they show that classical propositional logic can be given a paraconsistent characterization. In both cases, we (...) consider Hilbert-style systems as well as Gentzen-style sequent calculi and natural-deduction formalisms. (shrink)

Cover -- Half Title Page -- Title Page -- Copyright Page -- Original Title Page -- Original Copyright Page -- Contents -- Preface -- 1 Introduction -- 2 Probability And Possibility For Choice -- 1 Introductory -- 2 A Theory About Personal Power -- 3 A Criticism Of Keynes -- 4 Some More Theories About Personal Power -- 5 An Analogy Between Two Kinds Of Possibility -- 3 Probability And Natural Powers -- 1 Introductory -- 2 The Relation Between Epistemic (...) And Natural Possibility -- 3 A Criticism Of The Doctrine That 'Probable' Is Ambiguous -- 4 A Comparison Of Possibility With Probability -- 4 Some Unobservable Properties -- 1 Introductory -- 2 Certainty, Necessity And Hume -- 3 Epistemology And Determinism -- 4 An Analysis Proposed And Defended -- 5 Power And Law -- 5 Some Puzzles About Potentiality -- 1 Introductory -- 2 The idea of a 'circumstance' -- 3 The Distinction Between Intrinsic And Extrinsic Properties -- 4 The Refutation Of Actualism -- 5 The Refutation Of The Theory That 'All Power Is Conditional' -- 6 The Powers Of People Andthe Powers Of Things -- 1 Introductory -- 2 The Distinction Between Natural And Personal Power -- 3 The Refutation Of Ryle's Dispositionalist Account Of Human Capacities -- 7 IFS And Cans -- 1 Introductory -- 2 The Refutation Of The Theory That All Personal Power is Conditional -- 3 A Refutation Of The Orthodox Account Of 'I Can', As Equivalent To A Conditional Statement -- 4 A Contradiction In Dispositionalism -- 5 Further Objections To The Orthodox View -- 6 The Verification Of 'I Can': More About Trying -- 8 Deliberation, Freedom And Meaning -- 1 Introductory -- 2 More About Possibility And The Context Of Deliberation -- And About The Relation Between Power And Will -- 3 On Not Being Able To Help It: A: Discussion Of The Relationship Between Power And Responsibility -- 4 On The Meaning Of Words Like 'Can'. (shrink)

Hybrid deduction-refutation systems are presented for four first degree entailment based logics. The hybrid systems are shown to deductively and refutationally sound and complete with respect to their logics. The proofs of completeness are presented in a uniform way. This paper builds on work in [6], where Goranko presented a deductively and refutationally sound and complete hybrid system for classical logic.

Refutation systems are formal systems for inferring the falsity of formulae. These systems can, in particular, be used to syntactically characterise logics. In this paper, we explore the close connection between refutation systems and admissible rules. We develop technical machinery to construct refutation systems, employing techniques from the study of admissible rules. Concretely, we provide a refutation system for the intermediate logics of bounded branching, known as the Gabbay–de Jongh logics. We show that this gives a (...) characterisation of these logics in terms of their admissible rules. To illustrate the technique, we also provide a refutation system for Medvedev’s logic. (shrink)

Dissoi Logoi 4.6 presents a beautiful self-refutation argument, which I analyse here, offering a different assessment of its relation to self-contradiction and the Liar paradox from the only one available in the literature.

Perhaps everyone who can think has the concept of possibility, but no one understands it. The metaphysical theory of Determinism is a symptom of this lack of understanding, and the inconclusiveness of its opponents’ arguments indicates that the lack is universal. In this book, first published in 1968, the author shows that there are a number of different kinds on non-logical possibility, subtly interrelated, each requiring separate explanation. An original contribution to the subject, it is essential reading for all students (...) of philosophy. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of (...) mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their (...) relation such that classical logic is simulated where proofs and refutations are conclusive. (shrink)

LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.

Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cut-free’ sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and (...) past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. (shrink)

In Nos. 105 and 121 of the Kuang-ming Daily "Philosophical Supplement," there were two articles by Ch'ieh Ta-yu: "Some Opinions on the Problem of Determining the Object of Dialectical Logic" and "The Marxist Dialectical Method and Dialectical Logic." The first is quite general, while the second "elucidates the connections and differences" between dialectics and dialectical logic discussed in the first article. Almost all the basic points in both articles are erroneous. Actually, the author waves the banner of (...) antidogmatism and expounds a revisionist point of view. We will now refute his contentions. (shrink)

Luca Castagnoli, Ancient Self-Refutation. The Logic and History of the Self- Refutation Argument from Democritus to Augustine, Cambridge: Cambridge University Press 2010, pp. XX+394. Hardback, ISBN 9780521896313. In his book Ancient Self-Refutation L. Castagnoli rightly observes that selfrefutation is not falsification; it overturns the act of assertion but does not prove that the content of the act is false. He argues against the widely spread belief that Sextus Empiricus accepted the self-refutation of his own expressions. (...) Castagnoli also claims that Sextus was effective in answering to the self-refutation charge. The achievement of the book is the discovery that in passages where Sextus seems to embrace the selfrefutation of his expressions (PH 1.14-15), he does not use the term peritropé, technical for self-refutation, but the term perigraphé, which means self-bracketing. Selfbracketing is weakening one’s own thesis, but not overturning it. Castagnoli claims that Sextus embraces the self-bracketing of his expressions, but never accepts their selfrefutation. However, Castagnoli is not right in saying that self-refutation is a shameful mistake for everybody. The mature skeptic cannot even think that self-refutation is wrong, because it would be a dogmatic view. Sextus seems to accept self-refutation at the end of Against the Logicians, where he presents the argument against proof and the metaphor of the ladder (M 8.480-1). Regardless of Sextus’ declarations, we have reason to think that he does not avoid self-refutation in a pragmatic sense. Self-bracketing of his position is not a consistent dialectical strategy, as Castagnoli writes, but the end of a rational discussion. Sextus avoids absolute self-refutation (we cannot falsify what he suggests), but he is unable to avoid pragmatic self-refutation (there is no way to assert his position without contradiction). This is the case even if Sextus refuses to assert his position. (shrink)

A sufficient condition for an extension of positive logic with strong negation to be characterized by a class of finite trees is given.

A general theory of refutation systems is given. Some applications (concerning maximality and minimality in lattices of logics) are also discussed.

The goal of this paper is to generalize specific techniques connected with refutation rules involving certain normal forms. In particular, a method of axiomatizing both a logic L and its complement −L is introduced.

The beauty of Anselm’s ontological argument is, I believe, that no matter how one approaches it, one cannot refute it without making a significant metaphysical assumption, one that is likely to be contentious in its own right. Peter Millican disagrees. He introduces an objection according to which one can refute the argument merely by analysing its shallow logical details, without making any significant metaphysical assumption. He maintains, moreover, that his objection does not depend on a specific reading of the relevant (...) Anselmian text; in fact, Millican claims that his objection is applicable to every version of the ontological argument. In this paper, I argue that millican’s objection does not succeed, because, contrary to what he says, in order to justify his objection he does have to make a deep metaphysical assumption and rely on a specific reading of Anselm’s text. (shrink)